On the existence of critical compatible metrics on contact $3$-manifolds
Yoshihiko Mitsumatsu, Daniel Peralta-Salas, Radu Slobodeanu
TL;DR
The paper resolves the generalized Chern–Hamilton conjecture by classifying when a closed contact 3-manifold admits a critical compatible metric for the Chern–Hamilton energy. The authors derive a dichotomy: such a metric exists precisely when the underlying structure is Sasakian or when the Reeb flow is $C^\infty$-conjugate to an algebraic Anosov flow modeled on $\widetilde{SL}(2,\mathbb{R})$, leading to a complete topological classification of manifolds supporting critical metrics. They connect criticality to calibrated bi-contact structures and Anosov dynamics, proving sufficiency by constructing explicit metrics and proving necessity via a first-integral analysis of $\lambda^2=\text{(eigenvalue-squared of }h)$. Consequently, no contact structure on $\mathbb{T}^3$ admits a critical metric and overtwisted structures are ruled out; all critical metrics are global energy minimizers, with implications tied to Thurston geometries and algebraic Anosov behavior.
Abstract
We disprove the generalized Chern-Hamilton conjecture on the existence of critical compatible metrics on contact $3$-manifolds. More precisely, we show that a contact $3$-manifold $(M,α)$ admits a critical compatible metric for the Chern-Hamilton energy functional if and only if it is Sasakian or its associated Reeb flow is $C^\infty$-conjugate to an algebraic Anosov flow modeled on $\widetilde{SL}(2, \mathbb R)$. In particular, this yields a complete topological classification of compact $3$-manifolds that admit critical compatible metrics. As a corollary we prove that no contact structure on $\mathbb{T}^3$ admits a critical compatible metric and that critical compatible metrics can only occur when the contact structure is tight.